arXiv:2609.13917v1 Announce Type: new Abstract: Energy-Based Models (EBMs) provide a flexible framework for generative modeling by learning an energy landscape that assigns low energy values to reali...
arXiv:2410. 10137v5 Announce Type: replace Abstract: We develop Riemannian approaches to variational autoencoders (VAEs) for PDE-type ambient data with regularizing geometric latent dynamics, which we refer to as VAE-DLM, or VAEs with dynamical latent manifolds.
arXiv:2605. 27756v2 Announce Type: replace-cross Abstract: Linear dimensionality reduction methods such as proper orthogonal decomposition (POD) make high-dimensional data amenable to analysis by identifying the principal components, or modes, that capture the most variance, or energy, in the data and constructing a low-dimensional representation in the subspace they span.
arXiv:2501. 09876v3 Announce Type: replace-cross Abstract: Generative modeling aims to generate new data samples that resemble a given dataset.
arXiv:2510. 09468v3 Announce Type: replace Abstract: Latent manifolds of autoencoders provide low-dimensional representations of data, which can be studied from a geometric perspective.
The paper introduces a Nested Inductive Bias framework that uses a two‑stage diffeomorphic composition to pull back non‑Euclidean target geometries onto symmetric positive definite (SPD) manifolds. This approach allows the construction of curvature‑aligned Riemannian classifiers that respect both matrix constraints and the intrinsic relational geometry of data. Empirical results on kinematic, signal processing, and synthetic benchmarks show that class separability degrades when metric curvature does not match the data distribution, and the authors also propose the Rational Conformal Metric (RCM) for robust vectorized architectures.
arXiv:2601. 19179v2 Announce Type: replace Abstract: Autoencoders have long been considered a nonlinear extension of Principal Component Analysis (PCA).
arXiv:2603.13496v2 Announce Type: replace Abstract: Constructing reduced-order models (ROMs) capable of efficiently predicting the evolution of parameter-dependent high-dimensional dynamical systems...
The paper investigates how fine‑tuning pretrained visual encoders for faithful image reconstruction affects diffusion models that operate in the resulting latent space. It finds that such fine‑tuning reduces the effective dimensionality of the latent representation, causing standard velocity‑prediction flow‑matching to fit noise outside the low‑dimensional signal manifold and making optimization inefficient. Consequently, the authors propose using a clean‑data ($oldsymbol{x}_{0}$) parameterization, which focuses learning on the signal manifold and consistently improves text‑to‑image generation across multiple strong‑reconstruction encoders.
LocaLeJEPA extends Joint-Embedding Predictive Architectures (JEPA) by introducing a manifold-theoretic reformulation that aligns representation learning with non‑Euclidean geometric priors. It incorporates a locally Euclidean distribution‑matching regularizer, LocaLeReg, to stabilize training and scale to large models. Experiments on ImageNet‑1K show that LocaLeJEPA outperforms the original LeJEPA across both hyperbolic and Euclidean backbones, improving linear and k‑NN classification as well as downstream tasks.
arXiv:2607. 05653v1 Announce Type: new Abstract: Principal Component Analysis or PCA-like properties (orthogonality, variance ranking) are seldom realized in deep autoencoder architectures.
arXiv:2606. 07058v1 Announce Type: new Abstract: Variational autoencoders (VAEs) learn low-dimensional latent representations of high-dimensional data.